Conspiracy beliefs and perceptual inference in times of political uncertainty

Sociopolitical crises causing uncertainty have accumulated in recent years, providing fertile ground for the emergence of conspiracy ideations. Computational models constitute valuable tools for understanding the mechanisms at play in the formation and rigidification of these unshakeable beliefs. Here, the Circular Inference model was used to capture associations between changes in perceptual inference and the dynamics of conspiracy ideations in times of uncertainty. A bistable perception task and conspiracy belief assessment focused on major sociopolitical events were administered to large populations from three polarized countries. We show that when uncertainty peaks, an overweighting of sensory information is associated with conspiracy ideations. Progressively, this exploration strategy gives way to an exploitation strategy in which increased adherence to conspiracy theories is associated with the amplification of prior information. Overall, the Circular Inference model sheds new light on the possible mechanisms underlying the progressive strengthening of conspiracy theories when individuals face highly uncertain situations.

probabilistic processing could provide a complementary approach to addressing the existing link between CTs and uncertainty.
Surprisingly, few attempts have been made to investigate the potential links between perceptual inference and conspiracy ideations in a controlled experimental setting.Nevertheless, some results from the CT literature appear to be compatible with a probabilistic formalism.Dagnall and colleagues 31 explored the link between CTs and a wide range of cognitive-perceptual factors.They showed that these factors, including hallucination proneness, often conceptualized as false inferences 32 , were associated with CTs.Additionally, conspiracy ideations were found to be associated with illusory visual pattern detection 27,33 , which is a phenomenon that has regularly been explored through the prism of Bayesian theory 34 .
Very few papers have directly fitted computational models to behavioral data in nonclinical samples, with some noticeable exceptions exploring paranoia and/or conspiracy ideations 35,36 .Purely theoretical papers have also confirmed that computational approaches could help to better understand the spread of extreme beliefs including CTs on simulated or social media data [37][38][39] .Crucially, a more personalized computational lens 40 and a study of CTs in their ecological environment 41,42 seem to be needed to decipher the respective contributions of sociopolitical factors and information weighting to CT emergence.
Thus, combining the strengths of normative and ecological research during uncertain societal crises appears to be necessary for establishing a bridge between CT and inference quantification.In the present paper, we utilized Circular inference (CI), a Bayesian framework that has been proven effective in capturing not only JTC in patients with psychosis 43,44 but also both perceptual 45 and cognitive 46 inferential suboptimality in nonclinical populations.These last results suggest that the CI framework could be suitable for capturing other variations from suboptimal inference in the general population.Based on this idea, we hypothesized that by fitting the CI model to a simple bistable task, we could benefit from using an ideal setup to challenge the potential links between (i) the inferential mechanisms at play under conditions of extreme uncertainty and (ii) the dynamics of conspiracy ideations in large populations exposed to natural sociopolitical stress.

Measuring multilevel inference before and after stressful political events
Because we assumed that periods of great sociopolitical uncertainty lead to significant increases in individual levels of distress and favor inferential biases such as conspiracy endorsements, we explored conspiratorial beliefs and perceptual stability around polarizing political events in three independent Western countries (see Fig. 1): the United States of America (US, 2020 presidential elections), the United Kingdom (UK, 2021 BREXIT implementation) and France (FR, 2022 presidential elections).At each time point, healthy participants were instructed to rate their level of distress related to the ongoing event in their own country (later referred to as political distress, see Methods and Supplementary Material section: Self-reported measures).

Necker cube experiment
At each time point, the 623 enrolled participants performed an online bistable perception task based on the Necker cube (NC).The interpretation of the two-dimensional NC projected from a three-dimensional space naturally alternates between two possible configurations: a seen from above (SFA), or a seen from below (SFB) Figure 1.A repeated-measures design framing stressful political events in 3 different countries.Conspiracy ideations, political distress and perceptual stability were measured in the same participants (n = 623) via an online procedure, before and after the occurrence of a polarizing political event in three Western countries (M stands for month): the 2020 presidential election in the United States of America (n = 212, US), BREXIT implementation in the United Kingdom (n = 225, UK) and the 2022 presidential elections in France (n = 186, FR).We used T1 and T2 measures in the main model, while T3 was used in control analyses (see Supplementary Material section: Controlling for experimental design biases).
www.nature.com/scientificreports/cube (Fig. 2a).A perceptual stability score, ranging from 0 to 1, was estimated at the participant level.This score corresponds to the probability of switching from one interpretation to the other (0 means total instability, while 1 reflects a perceptive rigidity where the participant only sees one interpretation of the two, see the Methods section).Assuming a universal mechanism at the roots of belief formation, we merged the 3 samples after ensuring their comparability in terms of perceptual stability at baseline (Table 1, Fig. 2b,c; see also Supplementary Material section: Controlling for experimental design biases).Importantly, perceptual stability was tested for in lab/ online within-subject reproducibility on a pilot independent sample before running the final online experiment Bland-Altman plot of the agreement between online and in-lab methods comparing stability scores obtained in each condition for the same participants (n = 16).The x-axis represents the average scores of the two methods.The y-axis represents the mean difference between online and in-lab stability scores.The limits of agreements (LoA, pink dotted lines) are defined as the mean difference computed on all participants (pink line) ± 1.96 s.d., and each dot represents a participant.As all participants are included in the LoA, the methods are considered to be in agreement and may be used interchangeably.

Conspiracy adherence measures
The participants were instructed to self-rate their level of adherence to CTs, completing the Generic Conspiracist Beliefs Scale (GCB, see Methods section) at each time step.Replicating previous findings, we showed that conspiracy ideations were not normally distributed across the tested participants (W = 0.954 ; p = 0.440e -12 , Fig. 3a, Fig. S1a), suggesting that only a subpart of the general population commonly endorses such beliefs.The distribution of the total GCB scores differed across the three samples (χ 2 = 31.5,p < 0.001, η 2 = 0.348e -07 ) despite a similar pattern across subscales (Fig. S1a-b, Table S1), notably demonstrating a common preoccupation for information control.

Stress correlates at baseline
We assume that some participants might adopt information-processing strategies that can reduce the uncertainty induced by the framed political event.Notably, we expect that the search for stability would translate into high levels of confidence measurable at different levels of processing, from perception to conspiracy beliefs.Since belief in CTs has been proposed to be a coping strategy able to reduce the stress elicited by uncertainty, we also expect an association between great levels of confidence and low levels of distress.We first checked for associations between political distress at baseline (i.e., when uncertainty peaked) and: (i) perceptual stability on the one hand, and (ii) conspiracy endorsement on the other hand (Fig. 4a).Political distress was found to be negatively linked with both levels of inference (p = 0.028, ρ = − 0.120 and p = 0.007, ρ = − 0.094 respectively).We further confirmed these findings by splitting the sample into two subsamples according to stress: (i) a 'low stress' (LS) and (ii) a 'high stress' group (HS).Comparing these two groups at baseline, we confirmed a significant difference in both stability (U = 41,385, p = 0.002, Cohen's d = 0.140, Fig. 4c) and GCB scores (U = 43,411, p = 0.023, Cohen's d = 0.110), such as the LS group scored higher in both.www.nature.com/scientificreports/Then, we examined the influence of age, education, country, political distress and perceptual stability on GCB scores (F(6,616) = 11.24,p < 0.001, adjusted R 2 = 0.090).We found that age (estimate = − 0.142, p = 0.006), education (estimate = − 1.35, p < 0.001) and country (p < 0.001) were significantly associated with CTs, which further confirmed that political distress (estimate = − 0.405, p = 0.008) is associated with conspiracy endorsement, even after we controlled for those sociodemographic factors.

Fitting the circular inference model
Perception can be conceptualized as an inferential process in which noisy or ambiguous sensory data are combined with prior beliefs using the Bayes theorem [46][47][48][49] .However, humans are frequently utilizing suboptimal probabilistic reasoning.We previously suggested that deviations from "Bayes-optimality" could be caused by the circular inference (CI) phenomenon, i.e., internal amplification priors and sensory evidence through feedforward/feedback loops in brain circuits 50 .
To better understand the association between conspiracy theories and perceptual inference, we fitted a dynamical Circular Inference model to the Necker cube (NC) task 51 .When applied to this type of behavioral data, the CI model describes the process through which participants combine prior expectations about the visual appearance of three-dimensional (3D) objects and ambiguous visual input (such as illusory depth cues) to compute a 3D interpretation of the two-dimensional (2D) NC, as seen from above (SFA) or seen from below (SFB).
Figure 5 illustrates why a CI model may capture bistable perception more accurately than a Bayes-optimal model.Here, the two possible 3D configurations for the Necker cube (SFA or SFB) are represented as a single binary variable (Fig. 1a).The 3D configuration persists over time but with some volatilities, e.g., occasional switches from SFA to SFB, and vice versa.Low-level sensory features (contours, disparity, etc.) may support either SFA (positive sensory input) or SFB (negative sensory input).When the Necker cube was ambiguous, we modeled the sensory input as white noise, with a mean of zero and unit variance representing sensory noise.
We represent the subject's internal belief as a value that is positive or negative if SFA or SFB, respectively, is perceived with high confidence but near zero in the case of uncertainty (see Methods).In a Bayes-optimal model, the percept corresponds to a leaky integration of sensory inputs over time, and sensory gain and leak are determined by sensory precision and volatilities.This process can be conceptualized as sensory noise pushing around a ball situated in a bowl-shaped energy landscape (Fig. 5b).The percept (or most likely configuration) corresponds to SFA or SFB when the ball is situated on the right or left side of the bowl, respectively.The higher parts of this landscape correspond to high levels of confidence in the current percept, while the lower parts correspond to higher levels of uncertainty.Notably, because of volatility, the ball will spend more time in this region of uncertainty.Moreover, in the absence of sensory inputs to push it around, the ball will eventually fall to the bottom of the bowl (equiprobability of the SFA and SFB configurations).In other words, each percept loses its influence over time, and the probability of persistence decreases continuously in the absence of sensory input (which corresponds to the OFF durations of our NC behavioral task; Fig. 5d, dark yellow line).This first approach appears to contradict experimental observations (replicated in this study) that bistable perception is stabilized for longer "OFF" durations 52 .In contrast, in the presence of CI, sensory inputs are pushed in the direction of prior beliefs and influence future beliefs in turn.This will result in an amplification of the prior at the detriment of sensory evidence (Fig. 5a, blue arrow).Thus, the energy landscape becomes bimodal with two valleys that correspond to the SFA and SFB configurations (Fig. 5b).As a result, CI consistently generates stable and strong beliefs (the ball remains in the same valley for long periods), with sensory noise occasionally causing a perceptual switch (making the ball fall in the opposite valley).In the absence of sensory input, the ball falls to the bottom of the valley in which it is currently located and where it remains stuck.Thus, the influence of the previous percept does not decay over time, and perceptual switches become less likely to occur for longer OFF durations (Fig. 5d, green line), in better agreement with what has been previously observed at the behavioral level.Finally, SFA is perceived more frequently by most people and can be captured by different switching frequencies (volatilities) between SFA and SFB.This bias renders the SFA valley deeper and the "SFB" valley shallower and predicts a less stable or even an unstable SFB percept, as indeed observed in many subjects (see, for example, the second and fourth subjects from the top in Fig. 5e).
By using this approach, we could fit four model parameters contributing to the perceptual decision of each individual subject: the weight of the sensory gain (sensory), the amplification or prior beliefs (prior), the strength of the bias (bias), and a fourth parameter (penalty).This fourth parameter can be conceptualized as the reflection of an adaptive strategy that enables the perception of the less probable configuration to occur at least some of the time (see Methods).By using this model, we were able to capture a wide variety of responses (Fig. 5e).We checked whether these CI parameters could capture the effects of political distress and conspiracy adherence.Sensory weight was the only parameter positively associated with GCB scores at baseline (p = 0.030, ρ = 0.098, Fig. 4b), supporting the idea that participants more prone to CTs at baseline rely more on sensory evidence when asked to make a decision in a highly ambiguous environment.We confirmed this GCB-sensory weight association (estimate = 1.20, p = 0.051) even after controlling for the effects of age, education and political distress (F(4,618) = 11.86,p < 0.001, adjusted R 2 = 0.065).
To account for the heterogeneity in stress evolution, we split the sample into two subgroups according to their trajectories: a first subsample with decreasing stress (Dec, n = 330) and a second subsample with increased stress between T1 and T2 (Inc, n = 227; Fig. 6b).Considering that the Dec group should have adopted the most efficient coping strategies, we checked how the CI parameters and degree of conspiracy ideations changed over the same period in these two subsamples (Table S3).
A delta measure for each CI parameter was computed (parameter value at retest minus value at baseline), such as a positive delta indicated a gain in the parameter value, while a negative delta reflected a decrease in this parameter.The Dec group showed increased reliance on prior information in the bistable task between T1 and T2 (mean ΔPrior = 0.0811, s.d.= 1.11), while the Inc group showed decreased use of priors in the same period (mean ΔPrior = -0.121,s.d.= 1.14).This difference was statistically significant (t(460,65) = 2.07, p = 0.039, Cohen's d = 0.180 ; Fig. 6c-left).We found no differences in the 3 other CI parameters (Fig. S5).
We also computed a composite ΔGCB score corresponding to GCB at retest minus GCB at baseline, such that a positive delta corresponded to an increase in conspiracy adherence, while a negative delta resulted in a decrease.Because conspiracy ideations were proposed to act as a coping mechanism when facing uncertainty, we next ran an oriented test to test this hypothesis, which confirmed conspiracy strengthening among participants with decreased stress in comparison with that observed in the rest of the sample (t(429,52) = 1.65, p = 0.050, Cohen's d = 0.145; Fig. 6c-right).This finding supports a gain in the GCB score for the Dec group compared to the Inc  www.nature.com/scientificreports/ group.To confirm that the increase in GCB score was directly associated with an increase in Prior in the Dec group, we compared ΔPrior and ΔGCB in this specific subsample; these measures were found to be positively associated (p = 0.035, ρ = 0.116, Fig. 6d).

Discussion
A surge in CTs has been observed in recent years, and CTs have been proposed to act as coping strategies for the stress and perceived lack of control generated by global uncertainty [14][15][16][17][18] .CTs offer intuitive and easy-to-understand explanations to unsolved problems 53 .Links have already been established between conspiracy endorsement and some inference biases 6,[19][20][21] .However, very few studies have primarily focused on low-level perceptual aspects of conspiracy 27,28,31,33 , and limited efforts have been made to delve into the potential mechanisms of information processing that may convey such associations.
To address these concerns, we combined online assessments of bistable perception in large international samples with Bayesian modeling.By using this approach, we could quantify perceptual inference mechanisms and test their links with conspiracy ideations during periods of great sociopolitical uncertainty.We were able to capture the strengthening of conspiracy beliefs in nonclinical populations.Specifically, using the Circular Inference (CI) model, we highlighted a significant association between conspiracy endorsement and the overweighting of sensory information in the wake of political polarizing events, which was followed by a selective increase in prior reliance in those who subsequently decreased their stress levels.
Several attempts to model the features of conspiracy beliefs can be found in the literature.However, most of these models have either focused on the network scale 54 or remained purely theoretical, without experimental testing 40 .Recent findings have highlighted the added value of a computational framework to account for the emergence of conspiratorial beliefs during the COVID-19 pandemic 35 and the protective aspect of CTs against distress in a social context 55 .These studies used high-level cognitive tasks and focused mainly on paranoia, a condition that shares some phenomenological features with CTs but is also considered to be significantly different 56 , further justifying specific explorations.The quantitative approach proposed in the present work nicely completes these initiatives, adding to the testing of low-level inference, together with measurements of the emergence and strengthening of conspiracy beliefs.
In this study, we provide the first evidence for an association between sensory information overweighting in ambiguous contexts and a high level of conspiracy endorsement.This finding suggests that when uncertainty is assumed to peak, a subpart of the population that is more vulnerable to stress is prone to embracing conspiracy explanations based on intuitive reasoning.Motivated by the need to cope with uncertainty, these participants first adopt an "exploration" strategy, seeking explanations in their direct environment to inform their perceptual decisions.Interestingly, such a mechanism accounts for perceptual and inferential biases previously found to be associated with conspiracy ideations, such as illusory pattern detection 27,28,33 , aberrant salience attribution 22 , intuitive thinking 57,58 and the JTC phenomenon 23,24 .
We also explored the dynamic changes in model parameters after stress resolution by using a pre/post design surrounding the political events.We shed light on the association between prior knowledge amplification in perceptual decisions and the enhanced adherence to CTs in those who showed reduced stress level.This finding suggests that some participants coped with uncertainty by embracing conspiracy-oriented explanations, secondarily shifting to an "exploitation" strategy (Fig. S6), validating their newly established view and reinforcing their own beliefs.This second mode appears compatible with findings showing confirmation biases 59,60 and reality testing deficits 25 in people with CTs, making these beliefs more resilient to counterevidence.
These results can also be compared with models of the emergence and maintenance of clinical beliefs, such as delusional ideations.Indeed, prior research conceptualized delusion formation as the result of impaired associative learning processes driven by excessive prediction error 61 , a framework that was later extended to account for delusion persistence as aberrant reinforcement of previously learned associations 62 .Our results also add to previous work showing that parametric changes might mimic behaviors observed during the transition to psychosis 63 .It was shown using CI-based simulations that the seminal amplification of sensory information involved in the integration of aberrant causal relationships (during the transition to psychosis) subsequently constituted strong priors proposed as responsible for the stability of delusional contents from one psychotic episode to the next.Both approaches (predictive coding and Bayesian modeling) are congruent with (i) the idea that conspiracy endorsement is associated with the establishment of aberrant causal relations between random events 14 , and (ii) that conspiracy could be rooted in the self-reinforcement of previously integrated suboptimal beliefs.
While endorsing CTs may serve as an effective short-term coping strategy, it also appears to pave the way for the long-term strengthening of suboptimal beliefs (beliefs that would be computed through mechanisms deviating from Bayes' rule), making it maladaptive for stress regulation overall.Gaining a better understanding of this phenomenon has vast social implications.Humankind has experienced repeated periods of heightened uncertainty throughout history, ranging from civilizational collapses or wars to economic crises.In extending the well-established association between political distress and the endorsement of CTs 13 , our model also explains the recent rise in extremism and populism observed since the beginning of the twenty-first century in the global context of the pandemic, terror attacks and climate change.
We must acknowledge that this work has some limitations.First, although significant, some results exhibit small effect sizes (i.e., Cohen's d of approximately 0.2) and are not always replicated when countries are tested separately.Of note, small effect sizes were previously found to still have substantial significance when studies were conducted on large populations 64 .Importantly, small effects were expected because we attempted to capture an association between a low-level inference process (bistable perception) and a more complex cognitive process (conspiracy).However, these findings still constitute an important proof-of-concept demonstration that the CI model can capture small variations in nonclinical populations' perceptual decisions, paving the way for promising advancements in deepening our understanding of the mechanisms underlying belief rigidification.
A second limitation is that we cannot rule out that some participants may have felt hesitant in honestly reporting their views about CTs, due to the controversy and potential stigma surrounding conspiracy thinking.However, we think that our experimental design offers two advantages in the valid assessment of conspiracy endorsement.First, its online nature ensured anonymity and encouraged freedom of speech, as frequently observed on the internet and digital social media.Second, the joint use of a low-level perceptual task, the NC, provided access to a proxy of inference processing that is rarely prone to social biases, such as interviewer compliance.
A third limitation is the representativity of the sample: we indeed chose to recruit participants from three Western educated countries that are known for their high degree of polarization 65 .Although our sample may not represent the world population and various sociocultural factors can influence conspiracy adherence, we argue that the phenomenon under investigation follows some universal rules.First, links between sociopolitical uncertainty and the resurgence of conspiracy beliefs have already been observed at various times and locations, dating back to the Roman Empire 66 .Second, while the GCB total scores were distributed differently among our three samples (Fig. S1a), their qualitative distribution across GCB subscales followed the same pattern (Fig. S1b).Third, the pattern of associations between political distress and inference processing measures (perceptual stability and CI parameters) appears to be consistent across the three samples when tested separately (Fig. S2).
For the same reasons, we focused on the level of distress related to specific political events in the countries where we conducted the tests.Importantly, we did not consider other types of individual stress levels.Instead, we concentrated on the broader phenomenon of sociopolitical uncertainty.Similarly, our procedure did not allow us to have a direct measure of this uncertainty, which could constitute an interesting addition in future studies.Finally, while we observe an increase in GCB scores in some subpopulations, this phenomenon, referred to as "CTs strengthening, " could be due to (i) a widening range of CTs rated as believable across time (scoring on more items of the scale) or (ii) a strengthening of conviction (scoring higher on the same items).
Overall, this study highlights the potential of the Circular Inference model in examining subtle variations in inference processing associated with high-level cognitive beliefs.This model has already proven effective in accounting for the positive symptoms of schizophrenia 43,44,67 and schizotypal traits 46 ; however, this breakthrough opens up new avenues for applying quantitative approaches to dynamically explore subjective beliefs in nonclinical populations.By applying this computational framework, we delved deeper into the mechanisms underlying the emergence and maintenance of conspiracy beliefs, shedding light on their societal impact and providing insights that could be valuable for developing interventions aimed to counter the influence of CTs during highly uncertain periods.

Participants
Three independent samples were recruited using the Prolific © web-platform: 212 US citizens, 225 British citizens and 186 French citizens.The same protocol was administered 1 month before and 1 month after a major stressful political event: the 2020 US presidential election, the 2021 UK BREXIT implementation and the 2022 French presidential election (Fig. 1).The targeted participants were aged between 18 and 60 and had normal-to-corrected vision.They were from the nationality of the country of interest for each sample and regularly used social media.The exclusion criteria were a history of psychiatric or neurological disorder, strabismus, or eye surgery.From the initial sample (N = 755), 30 participants were excluded based on failed attentional checks (see Supplementary Material section: Controlling for experimental biases) or low reaction times (mean reaction time < 300 ms), while 102 were lost longitudinally.
The Prolific © web-platform (https:// www.proli fic.co/) ensures data privacy following standards of the European and UK data protection law (i.e., General Data Protection Regulation (GDPR), transposed into UK law as the UK GDPR).Informed consent was obtained from all participants and their sociodemographic characteristics were associated with their respective behavioural data through an anonymous ID randomly assigned at enrollment.This online study was approved by the ethics committee Comité de Protection des Personnes Nord-Ouest IV, and its methods complied with French regulations and were carried out in accordance with relevant guidelines.

Apparatus
The protocol was implemented in PsychoPy v.3, exported and hosted online on the Pavlovia.orgplatform.For the perceptual part of the experiment, participants were instructed to stand in total darkness, approximately 60 cm away from the screen and adjust it to be perpendicular to the floor with their eyes aligned to the fixation cross displayed at the centre of the screen.The NC task and the self-reported assessment of beliefs were administered in a randomized order (see also Supplementary Material section: Controlling for experimental biases).

The Necker Cube Task
Stimuli Visual stimuli representing Necker cubes (NC) were displayed in the centre of a black screen.The stimulus size was standardized across the participants using a matching method based on a standard credit card displayed on the screen that the participant was required to adjust in size before starting the experiment (See demo available at: https:// github.com/ Renau dJA/ Necker_ cube_ demo).

Procedure
The block-design of the task was inspired by Mamassian and Goutcher's 68 protocol.During each block, a NC was presented discontinuously.Referring to a forced-choice methodology, we asked participants to report their To predict individual responses, the model was tested on the same sequence of "OFF" duration that were used in each subject.We computed the pairwise statistics of two successive model responses (e.g.Probability of two successive SFA responses, SFA followed by SFB, SFB followed by a SFA, and two successive SFB, at each of the 7 delays, generating 28 measures).We then averaged the responses of 40 runs of the model (with identical OFF durations but different sensory noise samples).These predictions were compared with the pairwise statistics measured experimentally.
The CI model presented above could have up to 5 free parameters ( r on , r off , w , a , and P ), while the Bayes- optimal model would have a maximum of 4 free parameters ( a being fixed to 0).We reduced the number of free parameters in the CI model in the following way.Let us define a bias b (preference for SFA), such that r on = rb and r off = r(1 − b) , where r is the mean volatility.The temporal dynamics of the model are dominated by its effective loop strength, L St = a r .If L St < 1, the model acts as a leaky integrator, with "uncertainty" being the only stable state (Fig. 5c).On the other hand, if L St > 1, the model becomes bistable for moderate biases (Fig. 5b), while only SFA is stable for stronger biases (i.e., the "SFB-valley" becomes too shallow to trap the ball).While L St has a crucial influence on perceptual choice dynamics, r exerts only a moderate effect.Thus, we reduced the number of parameters by fixing it to 10 Hz for all subjects, corresponding to a mean sensory integration time constant of 100 ms.Thus, the 4 parameters of the CI model were the bias b , sensory gain w , loop strength for prior information L St and penalty term P.
While model comparison was not the main purpose of this study, we also tested a Bayes-optimal model (e.g., with a = 0) with the same degree of freedom using the same methods, keeping b, w, r and P as its four free parameters.Indeed, without prior amplification, persistence following a long OFF duration is achieved only for very low volatilities ( (a * r) < 1 OFF−duration ).We tested this model on 220 subjects extracted from the preelection American dataset.Our findings reveal a notable distinction in its ability to account for individual responses, as evidenced by the mean squared error (MSE) metrics (mean(MSE ci ) = 0.075, mean(MSE bayes ) = 0.093).This difference was statistically significant according to the Wilcoxon test, p = 2.673e -2 .Furthermore, the Bayesian information criterion (BIC) values corroborate these results, with BIC ci = − 549 and BIC bayes = − 502.

Model fitting procedure
For each model, we used MATLAB patternsearch to minimize the Euclidian distance between the predicted and measured pairwise response statistics for each OFF duration.For each subject and each session, patternsearch was repeated 100 times with different starting points, and the best parameter set was retained as the best model fit.We also performed parameter recovery and measured test-retest consistency between parameters measured in the same subject using data from the T2 and T3 time points (see Supplementary Materials and Fig. S4).

Data analysis and statistics
Characteristics of conspiracy adherence.The normality of the distributions was tested using the Shapiro-Wilk test.If the data were not normally distributed, further analyses were performed using nonparametric statistics.We compared GCB scores between males and females using a Mann-Whitney test, while GCB scores among the three US-UK-FR samples, across ISCED levels of education and across different age groups were compared using Welch ANOVAs.
The correlates of stress at baseline.We conducted a series of model-free analyses to confirm the association between political distress, stability score, and GCB.Again, due to the non-normal distribution of the GCB scores, we referred to Spearman rank correlations to explore linear associations, corrected for multiple comparisons based on the false discovery rate (FDR) method.These analyses were conducted on the whole sample, and on subsamples generated through a median split on the political distress score: the 'low stress' (LS, n = 310) and 'high stress' (HS, n = 313) subgroups.We used Mann-Whitney tests to assess the difference between these two subgroups regarding stability scores or GCB scores.We also used a linear regression model to confirm the association between political distress and GCB, adding age, education level and country as covariates to control for the effect of these sociodemographic factors.
Changes after political event resolution.We assessed the evolution of political distress, stability scores and GCB scores over time using Wilcoxon signed-rank tests for repeated measures.We then split our sample into two groups: Dec and Inc comprising individuals who showed decreased or increased stress, respectively, between the two time points.We computed a delta measure for each parameter that corresponded to the parameter's value at retest minus that at baseline.A positive value indicated a gain in the parameter, while a negative value indicated a decrease.Due to the normal shape of distributions in these composite scores and our sample size, we referred to Welch tests for group comparisons.
The same procedure was used to compare the two groups regarding the gain in GCB (ΔGCB).We performed a Welch's test for the oriented hypothesis that the Dec subsample would significantly increase its GCB score compared with the Inc subsample.Finally, a Spearman correlation test was used to check for an association between ΔAlpha and ΔGCB in the Dec subgroup.

Figure 2 .
Figure 2. The Necker cube (NC) task: procedure and validity.(a) The experimental procedure consisted of serial NC presentations.Each trial was decomposed into three steps (see Methods section).After a fixation cross of pseudorandomized duration (ISI) (1), the Necker cube was presented (2) until participants reported their interpretation of the stimulus: 'seen from above' (SFA) or 'seen from below' (SFB), using the right or left arrow of their keyboard, respectively (3).(b) Perceptual stability as a function of the interstimulus interval (ISI) for each national sample.US (United States of America, mean Stability Score = .587,s.d.= .172),UK (United Kingdom, mean Stability Score = .570,s.d.= .176)and FR (France, mean Stability Score = .565,s.d.= .1472).(c) Averaged stability scores at each time point for the three national samples.(d) Perceptual stability as a function of ISI, for online (mean stability score = .441,s.d.= .190)and in-lab methods (mean stability score = .500,s.d.= .140).(e)Bland-Altman plot of the agreement between online and in-lab methods comparing stability scores obtained in each condition for the same participants (n = 16).The x-axis represents the average scores of the two methods.The y-axis represents the mean difference between online and in-lab stability scores.The limits of agreements (LoA, pink dotted lines) are defined as the mean difference computed on all participants (pink line) ± 1.96 s.d., and each dot represents a participant.As all participants are included in the LoA, the methods are considered to be in agreement and may be used interchangeably.

Figure 5 .
Figure 5.The circular inference model.(a) The Circular Inference (CI) model relies on 4 complementary parameters described as follows: (1) the overall gain based on the sensory inputs ( w ), (2) an amplification of prior beliefs due to the presence of descending loops ( L St ), (3) a ON/OFF switch penalty causing volatility in percept generation ( P ), and (4) the configuration preference, or bias for the dominant percept ( r on − r off ) (see also Methods).(b,c) present the energy landscapes of the model, with and without prior amplification.(b) In the presence of noise, the prior amplification generates a bistable attractor allowing for a stabilization of the 3D configuration.SFA corresponds to the « seen from above » configuration, while SFB corresponds to the « seen from below » one.(c) On the contrary, a model without prior amplification (i.e., Bayes-optimal case) meets difficulties to stabilize on a given percept.(d) represents these patterns by plotting the persistence probability (PP) for both SFA and SFB.While the simple Bayes model exhibits a fall in the PP when OFF duration increases, the CI model allows for stabilized percept.(e) Illustration of the CI ability to capture the between-subject variability in the Necker cube bistable task.The cube PP is plotted as a function of the OFF duration for 4 random participants taken from the study.Circles correspond to experimental data and lines to the model fitting.